Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

We examine the following degenerate elliptic system:\begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v>0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!>\!0. $\end{document}We prove that the system has no stable solution provided \begin{document}$ p, \theta >0 $\end{document} and \begin{document}$ N_s: = N_1+(1+s)N_2< 2 + \alpha + \beta, $\end{document} where\begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document}This result is an extension of some results in [ 15 ]. In particular, we establish a new integral estimate for \begin{document}$ u $\end{document} and \begin{document}$ v $\end{document} (see Proposition 1.1), which is crucial to deal with the case \begin{document}$ 0 < p < 1. $\end{document}